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Why didn't “Europe” support the byzantine empire against the turks?

Why didn't Catholic Europe support the Byzantine empire against the Turks? Or did they? How? Yes, there were some differences between the Catholic and Orthodox interpretation of the Bible but the Turks were Muslims and should have been considered much worse (c.f. the "reconquista"), shouldn't they? They co-operated during the Crusades and the European empires (Franks, Habsburg) shouldn't Catholic Europe have appreciated Byzantium as a buffer towards the Muslim world?